The expression (mr), is called the moment of inertia of the body with respect to the axis of suspension.

The moment of inertia of a body, with respect to any axis, is the algebraic sum of the products obtained by mul tiplying the mass of each elementary particle by the square of its distance from the axis.

The expression (mr), is called the moment of the mass, with respect to the axis of suspension.

The moment of the mass with respect to any axis, is the algebraic sum of the products obtained by multiplying the mass of each elementary particle by its distance from the axis.

From the principle of moments, this is equal to the moment of the entire mass, concentrated at the centre of gravity. Denote the mass, or (m), by M, the distance of its centre of gravity from the axis, by k, and we shall have,

Σ(mr) = Mk

Substituting this in Equation (100), we have,

(101.)

(102.)

That is, the distance from the axis of suspension to the axis of oscillation is equal to the moment of inertia, taken with respect to the axis of suspension, divided by the moment of the mass, taken with respect to the same axis.

Let the axis of oscillation be taken as an axis of suspension, and denote its distance from the new axis of oscillation by l'. The distances of a, b p, k, from 0, will be r, l - r', &c., and the distance GO will be 7 - k. From the principle just enunciated, we shall have,

1 =

Σ[m(l - r)2] ̧.
M(l-k)

Or, performing the operation of squaring and reducing,

But

(ml) - 22(mrl) + Σ(mr2)

=

M(l-k)

is constant, hence (ml) Σ(m) × 12 = MP, also, 22(mrl) = 2Σ(mr) × 1 = 2Mkl; from Equation (102) we have, (mr) = Mkl. Substituting these values in the preceding equation, we have,

Hence, it follows that the axes of suspension and oscillation are convertible; that is, if either be taken as the axis of suspension, the other will be the axis of oscillation, and the reverse.

This property of the compound pendulum has been employed to determine experimentally the length of the seconds pendulum, and the value of the force of gravity at different places on the surface of the earth.

A straight bar of iron CD, is provided with two knifeedge axes, A and B, of hardened steel, at right angles to the axis of the bar, and having their edges turned towards each other. These axes are so placed that their plane will pass through the axis of the bar. The pendulum thus constructed is suspended on horixontal plates of polished agate, and allowed to vibrate about each axis in turn till, by filing away one of the ends of the bar, the times of vibration about the two axes are made equal. The distance AB is then equal to the length of the equivalent simple pendulum; that is, of a simple pendulum which will vibrate in the same time as the bar about either axis.

D

Fig. 106.

To employ the pendulum thus adjusted to find the length of a simple seconds pendulum at any place, the pendulum is carefully suspended, and allowed to vibrate through a very small angle; the number of vibrations is counted, and the time occupied is carefully noted by means of a well-regulated chronometer. The entire time divided by the number of vibrations performed, gives the time of a single vibration. The distance between the axes is carefully measured by an accurate scale of equal parts, which gives the length of the corresponding simple pendulum. To find the length of the simple seconds pendulum, we then make use of Proportion (97), substituting in it for t' and l' the values just found, and for t, 1 second; the only remaining quantity in the proportion is 7, which may be found by solving the proportion. This value of 7 is the required length of the simple seconds pendulum at the place where the observation is made. In making the observations, a variety of precautions must be taken, and several corrections applied, the explanation of which does not fall within the scope of this treatise. It is only intended to point out the general method of proceeding. By a long series of carefully conducted experiments, it has been found that the length of a simple seconds pendulum in the Tower of London is 3.2616 ft., or 39.13921 in. By a similar course of proceeding, the length of the seconds pendulum has been determined for a great number of places on the earth's surface, at different latitudes, and from these results the corresponding values of the force of gravity at those points have been determined according to the following principle:

From Equation (95), which is, t

solving with respect to g, and making t = 1,

we find, by

g
= T2l.

From this equation the value of g may be found at different places, by simply substituting for 7 the length of the

seconds pendulum at those places. In this manner, the value of g is found for a great number of places in different latitudes, and from these values the form of the earth's surface may be computed.

It has been ascertained in this manner that if the force of gravity at any point on the earth's surface be denoted by g, the force of gravity at a point whose latitude is 45°, by g', and the latitude of the place where the force of gravity is g', by l, we shall have,

g = g'(1 − .002695 cos2l).

125.

PRACTICAL APPLICATIONS OF THE PENDULUM.

One of the most important of the applications of the pendulum is to regulating the motion of clocks. A clock consists of a train of wheelwork, the last wheel of the train connecting with the upper extremity of a pendulumrod by a piece of mechanism called an escapement. The wheelwork is maintained in motion by means of a descending weight, or by the elastic force of a coiled spring, and the wheels are so arranged that one tooth of the last wheel in the train escapes from the upper end of the pendulum-rod at each vibration of the pendulum, or at each beat. The number of beats is registered and rendered visible on a dial-plate by means of indices, called the hands of the clock.

On account of the expansion and contraction of the material of which the pendulum is composed, the length of the pendulum is liable to continual variation, which gives rise to an irregularity in the times of vibration of the pendulum. To obviate this inconvenience, and to render the times of vibration perfectly uniform, several ingenious devices have been resorted to, giving rise to what are called compensating pendulums. We shall indicate two of the most important of these combinations, observing that all of the remaining ones are nearly the same in principle, differing only in the modes of application.

Graham's Mercurial Pendulum.

126. GRAHAM's mercurial pendulum consists of a rod of steel about 42 inches long, branched towards its lower end, so as to embrace a cylindrical glass vessel 7 or 8 inches deep, and having 6.8 in. of this depth filled with mercury. The exact quantity of mercury being dependent on the weight and expansibility of the other parts of the pendulum, must be determined by experiment in each individual case. When the temperature increases, the steel rod is lengthened, and, at the same time, the mercury expanding, rises in the cylinder. When the temperature decreases, the steel bar is shortened, and the mercury falls in the cylinder. By a proper adjustment of the quantity of mercury, the effect of the lengthening or shortening of the rod is exactly counterbalanced by the rising or falling of the centre of gravity of the mercury, and the axis of oscillation is kept at an invariable distance from the axis of suspension.

Harrison's Gridiron Pendulum.

127. HARRISON's gridiron pendulum consists of five rods of steel and four of brass, placed alternately with each other, the middle rod, or that from which the bob is suspended, being of steel. These rods are connected by cross-pieces in such a manner that, whilst the expansion of the steel rods tends to elongate the pendulum, or lower the bob, the expansion of the brass rods tends to shorten the pendulum, or raise the bob. By duly proportioning the sizes and lengths of the bars, the axis of oscillation may be maintained, by the combination, at an invariable distance from the axis of suspension. From what has preceded, it follows that whenever the distance from the axis of oscillation to the axis of suspension remains invariable, the times of vibration must be absolutely equal at the same place. The pendulums just de

Fig. 107.